Point of Intersection of Tangents

When studying conic sections such as parabolas, ellipses, and hyperbolas, the concept of tangents is crucial. A tangent to a curve at a given point is a straight line that just touches the curve at that point. The point of intersection of tangents is the point where two such tangents meet. This concept is particularly important in the context of parabolas, where tangents can be constructed from a point outside the curve.

Understanding Tangents to a Parabola

A parabola is defined as the set of all points that are equidistant from a fixed point (the focus) and a fixed line (the directrix). The equation of a parabola with its vertex at the origin and axis of symmetry along the y-axis is given by:

$$ y^2 = 4ax $$

where ( a ) is the distance from the vertex to the focus.

Tangent Equation

The equation of the tangent to the parabola ( y^2 = 4ax ) at the point ( (x_1, y_1) ) is given by:

$$ yy_1 = 2a(x + x_1) $$

Alternatively, if the tangent is at the point ( (at^2, 2at) ), the equation can be written in terms of the parameter ( t ):

$$ y = tx - 2at^2 $$

Point of Intersection of Two Tangents

When two tangents are drawn to a parabola from a point outside it, they intersect the parabola at two distinct points. The line joining these two points of contact is called the chord of contact. The point of intersection of the two tangents is not on the parabola itself but lies on the chord of contact.

Formula for the Point of Intersection

If two tangents are drawn to the parabola ( y^2 = 4ax ) from an external point ( (x_0, y_0) ), and they touch the parabola at ( (at_1^2, 2at_1) ) and ( (at_2^2, 2at_2) ), the point of intersection ( (X, Y) ) can be found using the following formula:

$$ X = \frac{at_1^2 + at_2^2}{2} $$ $$ Y = \frac{2at_1 + 2at_2}{2} $$

Simplifying, we get:

$$ X = \frac{a(t_1^2 + t_2^2)}{2} $$ $$ Y = a(t_1 + t_2) $$

Table of Differences and Important Points

Aspect	Chord of Contact	Tangent
Definition	The line segment joining the points of contact of two tangents from an external point.	A line that touches the curve at exactly one point.
Equation (for parabola (y^2 = 4ax))	(T = 0) where (T: yy_0 = 2a(x + x_0))	(y = tx - 2at^2) or (yy_1 = 2a(x + x_1))
Point of Intersection	Lies on the chord of contact.	The point where the tangent touches the parabola.
Relation to Parabola	Does not lie on the parabola itself.	Is a point on the parabola.

Examples

Example 1: Finding the Point of Intersection

Given a parabola ( y^2 = 12x ), find the point of intersection of the tangents drawn from the point ( (9, 12) ).

Solution:

The general equation of the tangent to the parabola ( y^2 = 4ax ) in terms of the slope ( m ) is:

$$ y = mx + \frac{4a}{m} $$

For the parabola ( y^2 = 12x ), ( a = 3 ). The equation of the tangent from ( (9, 12) ) is:

$$ 12 = 9m + \frac{12}{m} $$

Solving for ( m ), we get two values, ( m_1 ) and ( m_2 ). Let's say ( m_1 = 1 ) and ( m_2 = 4 ) (for the sake of this example).

The points of contact are then found by substituting ( m_1 ) and ( m_2 ) into the tangent equation:

$$ y = mx + \frac{12}{m} $$

For ( m_1 = 1 ):

$$ y = x + 12 $$

For ( m_2 = 4 ):

$$ y = 4x + 3 $$

The point of intersection ( (X, Y) ) is the solution to the system of equations formed by these two tangents. Solving them, we find the point of intersection.

Example 2: Using the Formula

Find the point of intersection of tangents to the parabola ( y^2 = 8x ) that are drawn from the point ( (2, 8) ).

Solution:

First, we find the parameters ( t_1 ) and ( t_2 ) using the equation of the chord of contact from ( (2, 8) ):

$$ T: yy_0 = 2a(x + x_0) $$ $$ 8y = 16(x + 2) $$

Solving for ( y ), we get ( y = 2x + 4 ). This is the equation of the chord of contact.

Now, we use the formula for the point of intersection:

$$ X = \frac{a(t_1^2 + t_2^2)}{2} $$ $$ Y = a(t_1 + t_2) $$

We need to find ( t_1 ) and ( t_2 ) such that the points ( (at_1^2, 2at_1) ) and ( (at_2^2, 2at_2) ) lie on the chord of contact. This requires solving a system of equations, which can be done by substituting the points into the chord of contact equation and solving for ( t_1 ) and ( t_2 ).

Once ( t_1 ) and ( t_2 ) are found, we substitute them back into the formulas for ( X ) and ( Y ) to find the point of intersection.

Understanding the point of intersection of tangents is essential for solving problems related to tangents and normals of parabolas and other conic sections. It is a concept that frequently appears in calculus, analytic geometry, and various applications such as optics and mechanics.